OCL23 — [OCL-2.1 RTF] Transitive closure operator

Specification: Object Constraint Language, formal/06-05-01

Section: 7.6 Collection Operations

Summary:

The concept of transitive closure is very useful for specifying well-formedness constraints on binary relations.

For a binary relation r:A->A and elements x,y,z in A, r(x,y) is true when r relates x to y and r is transitive whenever for any x,y,z in A, r(x,y) and r(y,z) imply r(x,z). The transitive closure of a binary relation r:A->A is the smallest relation, ^r:A->A, that contains r and is transitive.

In the UML specification, the definition of Classifier::allParents() : Classifier[*] as “all the direct and indirect ancestors of a generalized Classifier” could be equivalently paraphrased as the transitive closure of the Classifier::parents() : Classifier[*] relation.

Specifying the notion of transitive closure for the OCL specification can be problematic as this notion refers to a meta-property not expressible in first order logic (i.e., the smallest relation satisfying a given property). However, the notion of transitive closure can be specified in the context of a particular API for evaluating OCL iterator expressions such as that of the Eclipse OCL implementation of the OCL2.0 specification:

http://help.eclipse.org/ganymede/index.jsp?topic=/org.eclipse.ocl.doc/references/overview/OCLOverview.html

http://dev.eclipse.org/viewcvs/index.cgi/org.eclipse.mdt/org.eclipse.ocl/plugins/org.eclipse.ocl/src/org/eclipse/ocl/internal/evaluation/IterationTemplateClosure.java?root=Modeling_Project&view=markup

A synopsis of the algorithm for computing the transitive closure as an OCL iterator expression is available here:

http://dev.eclipse.org/newslists/news.eclipse.modeling.mdt.ocl/msg01390.html

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